The particles in solid are closely packed and held together by strong intermolecular forces. The building constituents have fixed positions and can only oscillate about their mean positions. They have definite shape and definite volume. The density of solids is high and they have low compressibility.
1. Classes of Solids: Two types of solids are known:
(i) Amorphous solids,
(ii) Crystalline solids.
(i) Amorphous solids: In amorphous solids, the arrangement of building constituents is not regular but haphazard. They may have a short range order. Their melting points are not sharp. They are isotropic in nature, i.e., their properties such as mechanical strength, electrical conductivity, etc. are same in all directions. Examples: rubber, quartz glass, etc.
(ii) Crystalline solids: In crystalline solids, the arrangement of building constituents is regular throughout the entire three-dimensional network. A crystalline solid has sharp melting point and is anisotropic in nature i.e., some of their physical properties such as electrical resistance or refractive index show different values when measured along different directions in the same crystal. It has a definite geometrical shape with flat faces and sharp edges. Examples: sodium chloride, quartz, etc.
Table 1.1: Distinction between Crystalline and Amorphous Solids
Property | Crystalline Solids | Amorphous Solids |
Shape | Definite characteristics and geometrical shape. | Irregular shape. |
Melting point | Melt at a sharp and characteristic temperature. | Gradually soften over a range of temperature. |
Cleavage property | When cut with a sharp edged tool, they split into two pieces and the newly generated surfaces are plain and smooth. | When cut with a sharp edged tool, they cut into two pieces with irregular surfaces. |
Heat of fusion | They have a definite and characteristic heat of fusion. | They do not have a definite heat of fusion. |
Isotropy | Anisotropic in nature. | Isotropic in nature. |
Nature | True solids. | Pseudo solids or super cooled liquids. |
Order in arrangement of constituent particles | Long range order. | Only short range order. |
Table 1.2: Different Types of Solids
Type of Solid | Constituent Particles | Bonding/Attractive Forces | Examples | Physical Nature | Electrical Conductivity | Melting Point | |
1. | Molecular solids | ||||||
(i) Non-polar | Molecules | Dispersion or London forces | Ar, CCl4, H2, I2, CO2 | Soft | Insulator | Very low | |
(ii) Polar | Dipole-dipole interactions | HCl, SO2 | Soft | Insulator | Low | ||
(iii) Hydrogen bonded | Hydrogen bonding | H2O (ice) | Hard | Insulator | Low | ||
2. | Ionic solids | Ions | Coulombic or electrostatic | NaCl, MgO, ZnS, CaF2 | Hard but brittle | Insulators in solid state but conductors in molten state and in aqueous solutions | High |
3. | Metallic solids | Positive ions in a sea of delocalised electrons | Metallic bonding | Fe, Cu, Ag, Mg | Hard but malleable and ductile | Conductors in solid state as well as in molten state | Fairly high |
4. | Covalent or network solids | Atoms | Covalent bonding | SiO2 (quartz), SiC, C (diamond), AlN, | Hard | Insulators | Very high |
C (graphite) | Soft | Conductor (exception) | |||||
2. Space Lattice and Unit Cell
Space Lattice: It is the three-dimensional arrangement of identical points in the space which represent how the constituent particles (atoms, ions, molecules) are arranged in a crystal. Each particle is depicted as a point.
Unit Cell: A unit cell is the smallest portion of a space lattice which, when repeated in different directions, generates the entire lattice.
A unit cell is characterised by six parameters, i.e., axial angles a, b and g and axial lengths a, b and c. Thus, unit cell of a crystal possesses all the structural properties of a given crystal.
3. Crystal Systems: On the basis of the axial distances and the axial angles between the edges, the various crystals can be divided into seven systems. These are listed in Table 1.3.
Table 1.3: Seven Primitive Unit Cells and their Possible Variations as Centred Unit Cells
Crystal system | Possible variations | Axial distances or edge lengths | Axial angles | Examples |
Cubic | Primitive, Body-centred, Face-centred | a = b = c | a = b = g = 90° | NaCl, Zinc blende, Cu, KCl, Diamond |
Tetragonal | Primitive, Body-centred | a = b ≠ c | a = b = g = 90° | White tin, SnO2, TiO2, CaSO4 |
Orthorhombic or Rhombic | Primitive, Body-centred, Face-centred, End-centred | a ≠ b ≠ c | a = b = g = 90° | Rhombic sulphur, KNO3, BaSO4 |
Hexagonal | Primitive | a = b ≠ c | a = b = 90°, | Graphite, ZnO, CdS, Mg, Zn |
Rhombohedral or Trigonal | Primitive | a = b = c | a = b = g ≠ 90° | Calcite (CaCO3) HgS (cinnabar), ICl, As, Sb, Bi |
Monoclinic | Primitive, End-centred | a ≠ b ≠ c | a = g = 90°, | Monoclinic sulphur, PbCrO4, Na2SO4.10H2O |
Triclinic | Primitive | a ≠ b ≠ c | a ≠ b ≠ g ≠ 90° | K2Cr2O7, CuSO4.5H2O, H3BO3 |
There can be 14 different ways in which similar points can be arranged in a three-dimensional space. These are called Bravais lattices.
4. Number of Atoms in a Unit Cell
(i) An atom lying at the corner of a unit cell is shared equally by eight unit cells and therefore, only one-eighth (1/8) of an atom belongs to the given unit cell.
(ii) An atom present on an edge is distributed among the four unit cells, therefore only one-fourth (1/4) of an atom belongs to the given unit cell.
(iii) A face-centred atom is shared between two adjacent unit cells. Therefore, one-half (1/2) of an atom lies in each unit cell.
(iv) A body-centred atom belongs entirely to one unit cell since it is not shared by any other unit cell. Therefore, its contribution to the unit cell is one.
Applying above stated points, let us calculate the number of atoms in the different cubic unit cells.
Simple cubic: 8 (corner atoms) ×
Body-centred cubic: 8 (corner atoms) ×
Face-centred cubic: 8 (corner atoms) ×
Table 1.4: Number of Atoms per Unit Cell
Type of cell | Number of atoms at corners | Number of atoms in faces | Number of atoms in the body of cube | Total |
Simple or primitive cubic | 8 × | 0 | 0 | 1 |
Body-centred cubic (bcc) | 8 × | 0 | 1 × 1 = 1 | 2 |
Face-centred cubic (fcc) | 8 × | 6 × | 0 | 4 |
5. Density of Unit Cell: Suppose edge of a unit cell of a cubic crystal is a, d is the density of the substance and M is the molar mass, then in case of cubic crystal,
Mass of unit cell = Number of atoms in unit cell × Mass of each atom
= z × m
Mass of each atom (m) =
m =
Mass of unit cell =
Volume of a unit cell = a3
Therefore, density of the unit cell,
d =
d =
6. Other Parameters of a Cubic System
(a) Atomic radius: It is defined as half of the distance between nearest neighbouring atom in a crystal. It is expressed in terms of length of the edge (a) of unit cell of the crystal.
(i) Simple cubic structure (sc): Radius of atom ‘r’ =
(ii) Body-centred cubic structure (bcc): Radius of atom ‘r’ =
(iii) Face-centred cubic structure (fcc): Radius of atom ‘r’ =
(b) Coordination number: It is defined as the number of nearest neighbours that a particle has in a unit cell. It depends upon the structure of unit cell of the crystal.
(i) Simple cubic structure (sc): Coordination number (C.N.) = 6
(ii) Body-centred cubic structure (bcc): C.N. = 8
(iii) Face-centred cubic structure (fcc): C.N. = 12
7. Packing Efficiency
Packing efficiency is the percentage of total space filled by the particles.
Packing efficiency =
(a) Packing efficiency in simple cubic structures:
Let ‘a’ be the cube edge and ‘r’ the atomic radius.
As the particles touch each other along the edge, therefore a = 2r
Volume of the unit cell = a3
Since one atom is present in a unit cell, its volume
v =
\ Packing efficiency =
=
= 52.36% = 52.4%
Therefore, 52.4% of unit cell is occupied by atoms and the rest 47.6% is empty space.
(b) Packing efficiency in ccp and hcp structures: The efficiencies of both types of packing, ccp and hcp, are equally good since in both, atom spheres occupy equal fraction (74%) of the available volume. We shall now calculate the efficiency of packing in ccp structure. Let the unit cell length be ‘a’ and face diagonal be ‘b’ (represented as AC in Fig. 1.7). In this figure other sides are not shown for the sake of clarity.
In triangle ABC, ∠ABC is 90°, therefore,
AC2 = b2 = BC2 + AB2
= a2 + a2 = 2a2
\ b =
If r is the radius of the sphere, we find
b = 4r =
or a =
As ccp structure has 4 atoms per unit cell, therefore the total volume of 4 spheres (v) is = 4 ×
Total volume of the unit cell (V) = a3 =
Packing efficiency =
=
=
Therefore, 74% of unit cell is occupied by atoms and the rest 26% is empty space.
(c) Efficiency of packing in bcc structures: In this case the atom at the centre is in touch with other two atoms which are diagonally arranged (see Fig. 1.8). The spheres along the body diagonal are shown with solid boundaries.
In DEFD,
b2 = a2 + a2 = 2a2
\ b =
In DAFD,
c2 = a2 + b2 = a2 + 2a2 = 3a2
\ c =
The length of the body diagonal c is equal to 4r, r being the radius of the sphere (atom). As all the three spheres along the diagonal touch each other,
c = 4r
Therefore, c = 4r =
a =
As already calculated, the total number of atoms associated with a bcc unit cell is 2, the volume (v) is, therefore,
Volume of the unit cell (V) =
Packing efficiency =
Therefore, 68% of unit cell is occupied by atoms and the rest 32% is empty space.
8. Close Packing of Constituents
(a) Close packing in one dimension
There is only one way of arranging spheres in a one-dimensional close packed structure, that is to arrange them in a row and touching each other. In one-dimensional close packed arrangement, the coordination number is 2.
(b) Close packing in two dimensions
Two-dimensional close packed structure can be generated by stacking (placing) the rows of close packed spheres. This can be done in two different ways as shown in Figs. 1.10(a) and (b).
(i) Square close packing [Fig. 1.10(a)]
(ii) Hexagonal close packing [Fig. 1.10(b)]
(c) Close packing in three dimensions
(i) Hexagonal close packing (hcp): The first layer is formed utilizing maximum space, thus wasting minimum space. In every second row the particles occupy the depressions (also called voids) between the particles of the first row (Fig. 1.11). In the third row, the particles are vertically aligned with those in the first row giving AB AB AB... arrangement. This structure has hexagonal symmetry and is known as hexagonal close packing (hcp). This packing is more efficient and leaves small space which is unoccupied by spheres. In hcp arrangement, the coordination number is 12 and only 26% space is free. A single unit cell has 4 atoms.
(ii) Cubic close packing (ccp): Again, if we start with hexagonal layer of spheres and second layer of spheres is arranged by placing the spheres over the voids of the first layer, half of these holes can be filled by these spheres. Presume that spheres in the third layer are arranged to cover octahedral holes. This arrangement leaves third layer not resembling with either first or second layer, but fourth layer is similar to first, fifth layer to second, sixth to third and so on giving pattern ABCABCABC... . This arrangement has cubic symmetry and is known as cubic closed packed (ccp) arrangement. This is also called face-centred cubic (fcc) arrangement [Fig. 1.12(a) and (b)].
The free space available in this packing is 26% and coordination number is 12.
9. Voids or holes: The empty spaces left between closed packed spheres are called voids or holes.
Voids are of three types:
(a) Octahedral voids: This void is surrounded by six spheres and formed by a combination of two triangular voids of the first and second layer. There is one octahedral void per atom in a crystal. The radius ratio
(b) Tetrahedral voids: These voids are surrounded by four spheres which lie at the vertices of a regular tetrahedron. There are 2 tetrahedral voids per atom in a crystal and the radius ratio is 0.225.
(c) Trigonal voids: The void, enclosed by three spheres in contact is called a trigonal void. There are 8 trigonal voids per atom in crystal and the radius ratio is 0.155.
10. Locating Tetrahedral and Octahedral Voids: All closed packed structures have both octahedral and tetrahedral voids. In a ccp pattern, there is one octahedral void at the centre of body and 12 octahedral voids on each of the 12 edges of the cube. Each void on the edge is shared by four other unit cells.
Octahedral void at centre of cube = 1
Effective number of voids at edges = 12 ×
Total number of octahedral voids = 1 + 3 = 4
In ccp structure, there are 8 tetrahedral voids. These are located at the body diagonals, two on each body diagonal at one-fourth of the distance from each end.
11. Radius Ratio: For ionic solids, the ratio of the radius of cation to that of anion is called radius ratio.
Radius ratio =
12. Crystal Defects: The defects are basically irregularities in the arrangement of constituent particles. Broadly, crystal defects are of two types, namely, point defects and line defects. Point defects are the irregularities or deviations from ideal arrangement around a point or an atom in a crystalline substance, whereas the line defects are the irregularities or deviations from ideal arrangement in entire rows of lattice points. These irregularities are called crystal defects.
13. Point Defects
Interstitials: Atoms or ions which normally occupy voids in a crystal are called interstitials.
Vacancy: When one of the constituent particles is missing from the crystal lattice, this unoccupied position is called vacancy.
Point defects can be classified into three types:
(A) Stoichiometric defects, (B) Impurity defects, and (C) Non-stoichiometric defects.
(A) Stoichiometric Defects: The point defects that do not disturb the stoichiometry of the solid are called stoichiometric defects. They are also called intrinsic or thermodynamic defects. These are of two types, vacancy defects and interstitial defects.
(a) Vacancy defect: When some of the lattice sites are vacant, the crystal is said to have vacancy defect. It results in decrease in density of the substance. This defect can arise when a substance is heated.
(b) Interstitial defect: When some constituent particles (atoms or molecules) occupy an interstitial site, the crystal is said to have interstitial defect. Due to this defect the density of the substance increases.
Vacancy and interstitial defects are generally shown by non-ionic solids because ionic solids must always maintain electrical neutrality. Ionic solids show these defects as Schottky and Frenkel defects as explained below:
(i) Schottky defect: This defect arises when equal number of cations and anions are missing from the lattice. It is a common defect in ionic compounds of high coordination number where both cations and anions are of the same size, e.g., KCl, NaCl, KBr, etc. Due to this defect, density of crystal decreases and it begins to conduct electricity to a smaller extent [Fig. 1.16(a)].
(ii) Frenkel defect: This defect arises when some of the ions of the lattice occupy interstitial sites leaving lattice sites vacant. This defect is generally found in ionic crystals where anion is much larger in size than the cation, e.g., AgBr, ZnS, etc. Due to this defect density does not change, electrical conductivity increases to a small extent and there is no change in overall chemical composition of the crystal [Fig. 1.16(b)].
(B) Impurity Defects: These defects arise when foreign atoms or ions are present in the lattice site (substitutional solid solutions) or in the interstitial sites (interstitial solid solutions). For example, when molten NaCl containing a little amount of SrCl2 is crystallised, some of the sites of Na+ ions are occupied by Sr2+. Each Sr2+ replaces two Na+ ions. It occupies the site of one ion and the other site remains vacant. The cationic vacancies thus produced are equal in number to that of Sr2+ ions.
(C) Non-stoichiometric Defects: These defects arise when stoichiometry of a substance is disturbed. These are of two types.
(a) Metal excess defect: This may occur in either of the following two ways:
(i) Metal excess defect due to anion vacancies: In this defect a negative ion from the crystal lattice may be missing from its lattice site leaving a hole or vacancy which is occupied by the electron originally associated with the anion. In this way crystal remains neutral. Alkali halides like NaCl and KCl show this type of defect.
F-centres: These are the anionic sites occupied by unpaired electrons. F-centres impart colour to crystals. They impart yellow colour to NaCl crystals, violet colour to KCl crystals and pink colour to LiCl crystals. The colour results by the excitation of electrons when they absorb energy from the visible light falling on the crystal.
(ii) Metal excess defect due to interstitial cation: In this defect an extra positive ion occupies interstitial position in the lattice and the free electron is trapped in the vicinity of this interstitial cation. In this way crystal remains neutral. For example, zinc oxide on heating loses oxygen and turns yellow.
ZnO
The excess of Zn2+ ions move to interstitial sites and the electrons to neighbouring interstitial sites.
(b) Metal deficiency defect: This type of defect generally occurs when metal shows variable valency. The defect arises due to the missing of cation from its lattice site and the presence of the cation having higher charge in the adjacent lattice site. For example, FexO, where x = 0.93 to 0.96.
14. (a) 13-15 Compounds: When the solid state materials are produced by combination of elements of groups 13 and 15, the compounds thus obtained are called 13-15 compounds. For example, InSb, AlP, GaAs, etc.
(b) 12-16 Compounds: Combination of elements of groups 12 and 16 yield some solid compounds which are referred to as 12-16 compounds. For example, ZnS, CdS, CdSe, HgTe, etc. In these compounds, the bonds have ionic character.
15. Magnetic Moments: The magnetic properties of substances result from their magnetic moments associated with individual electrons. Each electron has a magnetic moment, origin of which lies in two sources. It is a known fact that an electron shows two types of motions, i.e., it rotates (spins) around its own axis and simultaneously revolves around the nucleus (orbital motion). An electron in motion is just like a small current loop. Two types of motions give rise to two types of magnetic moments—spin and orbital magnetic moments. Spin moment is directed along the spin axis and is shown in up or down direction [Fig. 1.19(b)]. Orbital motion also generates a magnetic field and thus gives rise to orbital moment along its axis of rotation [Fig. 1.19(a)]. In this way each electron of the atom behaves like a small bar magnet having permanent orbital and spin magnetic moments. Magnetic moments are measured in Bohr magneton (µB) unit (B.M.).
1 B.M. =
where e is charge on electron; h is Planck’s constant; m is the mass of electron and c is the velocity of light.
Depending upon two spin motions (clockwise and anticlockwise), spin magnetic moment may acquire two values
16. Magnetic Properties of Solids: On the basis of their magnetic properties, substances can be classified into five categories.
(a) Diamagnetic: Diamagnetic substances are weakly repelled by the external magnetic field. The atoms of these substances have all paired up electrons. As pairing of electrons cancel their magnetic moments, they lose their magnetic character. NaCl, H2O, TiO2 and C6H6 are some examples of diamagnetic substances.
(b) Paramagnetic: Paramagnetic substances are weakly attracted by the external magnetic field. The atoms of these substances have one or more unpaired electrons. Paramagnetism is temporary and is present as long as external magnetic field is present. O2, Fe3+, Cr3+, TiO, VO2, Cu2+ are some examples of paramagnetic substances.
(c) Ferromagnetic: Ferromagnetic substances are strongly attracted by the external magnetic field. In solid state, the metal ions of these substances are grouped together into small regions called domains. Each domain acts as a tiny magnet. When such a substance is placed in a magnetic field all the domains get oriented in the direction of magnetic field and a strong magnetic effect is produced. This ordering of domains persists even when the magnetic field is removed and the ferromagnetic substance becomes a permanent magnet. Thus, besides strong attractions, these substances can be permanently magnetised.
(d) Antiferromagnetic: These substances have domain structure similar to that of ferromagnetic substances but their domains are oppositely oriented and cancel out each other’s magnetic moment. MnO is an antiferromagnetic substance.
(e) Ferrimagnetic: In ferrimagnetic substances due to unequal number of magnetic moment in parallel and antiparallel directions, the net magnetic moment is small. These substances lose ferrimagnetism on heating and become paramagnetic. Fe3O4 and ferrites like MgFe2O4 and ZnFe2O4 are examples of such substances.
Curie Temperature: The temperature at which a ferromagnetic substance loses its ferromagnetism and attains paramagnetism only is called curie temperature. For iron, the curie temperature is 1033 K, for Ni it is 629 K and for Fe3O4 it is 850 K. Below this temperature paramagnetic substances behave as ferromagnetic substances.
17. Electrical Properties: Solids are classified into three groups on the basis of their electrical conductivities:
(a) Conductors: These generally include metals. Their conductivity is of the order of 104–107 ohm–1 m–1.
(b) Semiconductors: Those solids which have intermediate conductivities ranging from 10–6 to 104 ohm–1 m–1 are classified as semiconductors. As the temperature rises there is a rise in conductivity because electrons from the valence band jump to conduction band.
(c) Insulators: These are solids which have very low conductivity values ranging from 10–20 to 10–10 ohm–1 m–1.
Causes of conductance in solids: In most of the solids conduction takes place due to migration of electrons under the influence of electric field. However, in ionic compounds, it is the ions that are responsible for the conducting behaviour due to their movement.
In metals, conductivity strongly depends upon the number of valence electrons available in an atom. A band is formed due to closeness of molecular orbitals which are formed from atomic orbitals.
If this band is partially filled or it overlaps the higher energy unoccupied conduction band, the electrons can flow easily under applied electric field and the solid behaves as conductor [Fig. 1.21(a)]. If the gap between valence band and next higher unoccupied conduction band is large, electrons cannot jump into it and such a substance behaves as insulator. [Fig. 1.21(b)]
If the gap between the valence band and conduction band is small, some electrons may jump from valence band to the conduction band. Such a substance shows some conductivity and it behaves as a semiconductor [Fig. 1.21(c)]. Electrical conductivity of semiconductors increases with increase in temperature, since more electrons can jump from valence to conduction band. Silicon and germanium show this type of behaviour and are called intrinsic semiconductors.
(d) Doping: It is a process by which impurity is introduced in semiconductors to enhance their conductivity.
n-type semiconductor: When silicon or germanium crystal is doped with a Group 15 element like P or As, the dopant atom forms four covalent bonds like a Si or Ge atom but the fifth electron, not used in bonding, becomes delocalised and contributes its share towards electrical conduction. Thus, silicon or germanium doped with P or As is called n-type semiconductor, n indicates negative charge of electron since it is the electron that conducts electricity [Fig. 1.22(b)].
p-type semiconductor: When silicon or germanium is doped with a group 13 element like B or Al, the dopant atom forms three covalent bonds, but at the place of fourth electron a hole is created. This hole moves through the crystal like a positive charge giving rise to electrical conductivity. Thus, Si or Ge doped with B or Al is called p-type semiconductor ( p stands for positive hole), since it is the positive hole that is responsible for conduction [Fig. 1.22(c)].
Diode: Diodes are made by the combination of n-type and p-type semiconductors. They are used as rectifiers.
Transistors: These are used to detect or amplify radio or audio signals. They consist of pnp or npn sandwich semiconductors.
Photodiode: These are diodes which are capable of converting light energy into electrical energy and are used in solar cells.
18. Structure of Some Ionic Solids
(a) Ionic solids of the type AB
(i) NaCl (fcc): Cl– in ccp, Na+ occupy all the octahedral voids.
Coordination number is 6 : 6 and
Number of formula units per unit cell = 4.
(ii) CsCl (bcc): Cl– ions at the corners of a cube, Cs+ ion at the body centre and vice-versa.
Coordination number is 8 : 8 and
Number of formula units per unit cell = 1.
(iii) ZnS (Zinc blende): S2– ions form ccp structure and Zn2+ ions occupy alternate tetrahedral voids, coordination number is 4 : 4 and
Number of formula units per unit cell = 4.
(iv) ZnS (Wurtzite): S2– ions form hcp structure and Zn2+ ions occupy alternate tetrahedral voids. Coordination number is 4 : 4 and
Number of formula units per unit cell = 4.
(b) Ionic solid of the type AB2
CaF2 (Fluorite): Ca2+ ions form ccp structure and F – ions occupy all tetrahedral voids. Coordination number is 8 : 4 and
Number of formula units per unit cell = 4.
(c) Ionic solid of the type A2B
Na2O (Antifluorite structure): O2– ions form ccp structure and Na+ ions occupy all tetrahedral voids. Coordination number is 4 : 8.
Important Formulae
1. Density of unit cell (d) =
2. Table 1.5: Different Parameters of Cubic System
Unit cell | No. of atoms per unit cell | Distance between nearest neighbour | C.N. | Radius |
Simple cubic | 1 | a | 6 | |
Face-centred cubic | 4 | 12 | ||
Body-centred cubic | 2 | 8 |
3. Packing efficiency =
Table 1.6: Packing efficiency of different crystals
S.No. | Crystal system | Packing efficiency |
(i) | Simple cubic | 52.4% |
(ii) | Body-centred cubic | 68% |
(iii) | Face-centred cubic | 74% |
(iv) | Hexagonal close-packed | 74% |
4. Radius ratio =
Table 1.7: Structural arrangement of different radius ratios of ionic solids
Radius ratio | Possible coordination number | Structural | Examples |
0.155 – 0.225 | 3 | Trigonal planar | B2O3 |
0.225 – 0.414 | 4 | Tetrahedral | ZnS, SiO44– |
0.414 – 0.732 | 6 | Octahedral | NaCl |
0.732 – 1.0 | 8 | Body-centred cubic | CsCl |
5. If R is the radius of the spheres in the close packed arrangement, then
(i) Radius of octahedral void, r = 0.414 R
(ii) Radius of tetrahedral void, r = 0.225 R
6. In a close packed arrangement:
(i) Number of octahedral voids = Number of atoms present in the close packed arrangement.
(ii) Number of tetrahedral voids = 2 × Number of atoms present in the close packed arrangement.
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